PAPR Problem for Walsh Systems and Related Problems

Abstract

High peak values of transmission signals in wireless communication systems lead to wasteful energy consumption and degradation of several transmission performances. We continue the theoretical contributions made by Boche and Farell toward the understanding of peak value reduction, using the strategy known as tone reservation for orthogonal transmission schemes. There it was shown that for orthogonal frequency-division multiplexing (OFDM) systems, the combinatorial object called arithmetic progression plays an important role in setting limitations for the applicability of the tone reservation method. In this paper, we show that the combinatorial object introduced as perfect Walsh sum (PWS) plays a similar role for code-division multiple access (CDMA) systems as arithmetic progression for OFDM systems. By specific construction, we show that for a chosen numbers $m$ and $n$ , all subsets $\mathcal {I} $ of the set $[N]$ of the first $N=2^{n}$ natural numbers, which has the density in $[N]$ larger than a given $\delta \in (0,1)$ , i.e., $\left |{ \mathcal {I} }\right |/N\geq \delta $ , and which is sufficiently large enough, in the sense that $\left |{ I }\right |\geq 2(2/\delta)^{2^{m}-1}$ , contains a PWS of size $2^{m}$ . By means of this result, and motivated by the previously mentioned connection between arithmetic progression and PWS, we show results for the PWS which are analogous to the famous Szemeredi theorem on arithmetic progressions, Conlon-Gower’s theorem on probabilistic construction of “sparse” sets containing an arithmetic progression, and even a solution of an analogon to the Erdős’ conjecture on arithmetic progressions. Those results give in particular an insight into the asymptotic limitations of tone reservation method for the CDMA systems. Besides, we show that a subset $I$ of $[N]$ is a PWS if and only if the embedding inequality of the subspace of $L^{1}([{0,1}])$ , containing linear combinations of Walsh functions indexed by elements of $\mathcal {I} $ , holds with the minimum possible embedding constant $\sqrt { \left |{ \mathcal {I} }\right |}$ . The corresponding approach based in particular by the fact that the PWSs are the only Walsh sums having unit $L^{1}$ -norm, proven in this paper. By means of that results, we show that the minimum possible threshold constant for which the tone reservation method is applicable yields $\sqrt { \left |{ \mathcal {I} }\right |}$ if and only if the information set $\mathcal {I} $ is a PWS.